Structure of some liquid transition metals using integral equation theory

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PRAMANA __ journal of physics

Structure of some liquid transition metals using integral equation theory

O AKINLADE, A M UMAR* and L A HUSSAIN t

Department of Physics, University of Agriculture, Abeokuta, Nigeria

*Department of Physics, Uthman dan Fodio University, Sokoto, Nigeria tDepartment of Physics, University of Ibadan, Ibadan, Nigeria

MS received 9 August 1996

Abstract. We present the results of calculations of the structure factor

S(q)

of some liquid 3d transition metals using the self consistent hybridized mean spherical approximation (HMSA) integral equation. The local pseudopotential used is composed of the empty core model and a part that takes care of s-d mixing through an inverse scattering approach to model the interionic pair potential. The results presented are in very good agreement with experiment for most of the systems investigated near freezing, as well as for the noble metals Cu, Ag and Au, thus, confirming the reliability of the pseudopotential in the present integral equation scheme.

Keywords. Liquid transition metals; integral equation theory.

PACS No. 61.25 1. Introduction

The central problem of liquid state physics is the determination of the pair distribution function

g(r)

or its Fourier transform, the static structure factor

S(q).

At least for a system described solely in terms of volume and pair forces, the knowledge of

g(r)

is all that we need to know. Among the standard statistical mechanics tools that can be used for this kind of study namely computer simulation, integral equations or perturbation calculations, the thermodynamically self consistent (TSC) integral equation approach has been very successful in recent times. Its success is mainly due to the technical advancement made in the development of two powerful and efficient numerical algorithms for solving nonlinear integral equations [1]. These techniques lead to a drastic reduction in the number of iterations necessary for convergence. Here we intend to adopt the hybridized mean spherical equation of Zerah and Hansen [2]

which interpolates between two older closures, the HNC (hypernetted chain) and the soft core mean spherical approximation (SMSA), by means of a switching constant chosen to enforce thermodynamic self consistency. The success of the HMSA is well established for one [3, 4] and two component systems [5].

The other important ingredient for such calculations is the interionic pair potential, usually obtained from a suitable pseudopotential. For transition metals unlike what happens for simple metals, there is not that large number of suitable pseudopotentials.

One of the most popular is that based on the first principle study made by Wills and 271

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Harrison (WH) [6] which uses separate treatments for the s-p and d states leading to an effective pair potential that takes into account the effect of s-d hybridization. Despite the success of the theory in the solid state, results for the structure factors of liquid 3d transition metals using molecular dynamics and other complicated liquid state theories have not been that reliable [7, 8]. The root cause of the problem lies in the fact that the position and depth of the first minimum of the WH potential are shifted towards short distances and progressively too deep. An important inference that can be deduced from these is that the liquid state properties apprears to be a severe test of the potential since a correct description is needed to obtain structural properties of such systems.

Essentially, for transition metals, one has to consider the fact that the tightly bound d electrons hybridize with the nearly free-electrons resulting in a partially filled d band, crossing the Fermi energy. The presence of the d band has been a serious impediment to the application of pseudopotential perturbation theory for such systems. Bretonnet and Silbert [9] have developed a pseudopotential which treats the sp states using an Ashcroft type empty core pseudopotential characterized by a core radius Re, and have made a suitable adjustment to incorporate the s-d mixing by an approximate potential inside the core. The relative success of the model can be deduced from its applications to perturbation calculations [10] and variational modified hypernetted chain integral equations [11]. On the basis of their calculation we feel that it would be worthwhile to look at all the liquid 3d transition metals and consider the modifications that may be necessary in applying the same theories to the noble metals.

The layout of the present paper is as follows. In § 2, we outline the theoretical basis of the work in terms of the pseudopotential used and the HMSA integral equations. In § 3, we compare our results for S(q) with those obtained experimentally in [13]. In §4 we present our conclusions.

2. Theory

2.1 Pseudopotential and interatomic pair potential

Given a system of particles in a volume D, and at a given density p = N / D , assumed to be interacting via a pairwise potential V(r), the pair potential can be expressed as

_ ~ [ [ ' ~ 2 ~ . . s i n q r , 7

V(r) = 1 - I - r N ~ q ) - - a q | , (1)

J o 7z q d

where FN(q) is the normalized energy wave-number characteristics and Z s represents the effective number of conduction electrons per ion. For transition metals, in order to account for hybridization, Z s takes on non-integer values [12]. The pseudopotential to be used [9] is constructed from the superposition of two potentials, one to account for the sp contribution and the other, for the d-band contribution taken together, these results in the following expression for the potential

T.2n=lBne(-r/na) r < R c

W ( r ) = _ z ~

(2)

r > R c r

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Integral equation theory

By Fourier transformation of eq. (2), the resulting form factor can be expressed as

B1H1

8B2H 2 x

w(q)=ana3p

( l + a 2 q 2 ) 2 4 ( 1 ~ 2 ) 2 ]

4 q Pcos(qRc).

(3) Usually, one requires that the potential and its first derivative be continuous at r = R c.

This enables us to define B 1 and B 2 as

Z s ( 1 2 a ' ~ [Rc'~

BX=Rc \

- ~ c J exp ~-~--), (4)

2Z s ( a

B 2 = -~-c \~--~c - 1 ) exp ( 2 ~ ) (5,

with X, =

n2a2q 2

and Y, =

Rc/na, H,

is given by

sin(qRc)

H n = 2 - e r' [ Y . ( 1 + X . ) - (1 - X n ) ] na---~

+ [2 + Y,(1 +

Xn)]Cos(qRc) 1,

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the parameter a being a measure of the softness of the repulsive potential. The dielectric screening function e(q) is given by

4he 2

e(q)

= 1 - q2 z(q)[1 -- G(q)]. (7)

x(q) is the Lindhard function and

G(q)

is the local field factor of Ichimaru and Utsumi [14]. FN(q) , the normalized energy-wavenumber characteristic can be written as

q2 2 1

,8,

In essence the potential

V(r)

is a function of three parameters: the core radius R c, the softness parameter a and

Z s,

the number of conducting electrons. We shall present the numerical values for the systems studied and from the equations above, the pair potential can be constructed.

2.2

Integral equation method

The pair correlation function

o(r)

of classical fluid interacting through a pair potential

• (r) is determined through the solution of the famous Ornstein-Zernike (OZ) equation. With ~(r) =

h(r) - c(r),

we have

7(r) = p f

^Ul)dV ⁽⁹⁾

which decomposes the total correlation function

h(r) = o(r)

- 1 into the direct correlation

c(r)

and the totality of the indirect effects mediated by third particles which itself is correlated to the original two. To solve the O Z integral equation, one needs a closure between h(r) and

c(r).

Here we intend to use the H M S A closure [2]. Implementation of

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this integral equation is as follows, first the potential

V(r)

is split into (at(r) and

(a~(r),

the short range repulsive and long range attractive terms respectively, according to

(at(r) = { V(r)- V(ro) r <~ r o

(10)

r ~ r o '

V(r°) r~r°

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(at(r) = ~ V(r) r >1

r o '

where

V(ro)

is the pair potential evaluated at its first minimum position r o. From (10), (1 i), Zerah and Hansen [2] proposed for g (r) an expression of the form

g(r) = e - ~ ' ( ' ) I 1 +

exp(f(r)[?(r)-fl(az(r)])-i l f ( r ) '

(12) wheref(r) is given by

f(r)

= 1 - e -~' (13)

and the parameter ~ is varied until full consistency between the virial and compressibility equation of state is achieved. As can be illustrated, the switching functions

f(r)

satisfies the limiting behaviour

lim f (r) = 0 (14)

r " * 0

so that (12) reduces in that limit to the soft-core mean spherical approximation closure, and the limiting behaviour

lim

f(r)

= 1 (15)

r ~ o o

corresponds to the HNC closure.

In order to obtain a, one follows the standard procedure of relating this adjustable parameter to achieve consistency between the relation arising from the compressibility route in the grand canonical ensemble, which is the inverse long-wavelength limit of the structure factor

[ ; I 1

Zr=tiP-1 l+4zcp h(r)r2dr -S(O)

and that obtained from the virial equation of state

(Z v) = Pt- ~ + Po + P1 +

e2, where

p o = P [ 2 n ~ n ) +

2 d2u(n)]

n -d--Tn2 j ,

/'co 2 [ - r 0 63 r 6 3

P2=2~zp2f:drr2g(r)[3f~-nf-~]2V(r),

(16)

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(18) (19) (20)

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Integral equation theory

where the superscript V in (17) refers to the virial pressure route. The number density p is related to the electronic density n by p = nZ, and u(n) is the structure-independent contribution to the energy, consisting of the ground-state electron gas energy, the average interaction between electrons and the noncoulombic part of the bare-ion pseudopotential. The evaluation of (17) could be very delicate and thus aiming only at statistical mechanics self-consistency for liquid structure at fixed density, we determine the parameter • by demanding equality between (16) and the expression to which (17) reduces when the density dependencies of u and V(r) are dropped. This procedure was adopted by Pastore and Kahl [15] and Lai et al [3] in their studies.

Essentially the value ofct obtained at a given thermodynamic state is used to evaluate g(r) in (12) from which the structure factor can be obtained by Fourier transformation.

3. Numerical results and discussion

Table 1 lists our input parameters for the systems investigated as well as the values of the consistency parameter ~. For a given ~, the system of equations (9) and (12) is solved, taking advantage of the combination of the Newton-Raphson and successive substitu- tion method as proposed by Gillan [1]. This we have done and in all cases we find that 2048 points with a step size Aria = 0.025 (a being the Wigner Seitz radius) are sufficient for our calculations.

The parameters Re, a and Z s characterizing the potentials used in the present calculations have been chosen thus; R c is fitted to reproduce numerically the observed isothermal compressibility. The parameter a is the softness parameter and has an influence on the softness of the potential V(r), which is restricted to the domain 4 < Rc/a < 5 [11, 16]. This results in a potential that has the required form, i.e. a short ranged repulsive part and an attractive tail. Any value outside this results in a purely repulsive form. For a, we have chosen a value that best reproduces the oscillations of g(r) for the system being investigated. We observe that this works quite well for 3d transition metals but for the 4d and 5d noble metals Ag and Au, the ratio Rc/a is in the range 5.0 to 5"5.

The last parameter is Z s and following the observation of Moriarty [17] that for 3d and 4d transition metals, Z s takes on values in the narrow range 1.1 < Z s < 1.7 with a typical value of 1.4. We have used the typical value for all the systems studied here.

Table 1. Temperature (K), atomic volume (t)), pseudopotential parameters a and R c and consistency parameter cc We note that in all cases Z s = 1.4.

Element T(K) ~(a.u) a ( a . u ) Rc(a.u) Sc 1812 144.90 0.361 1.63 0"346 Ti 1933 130.38 0.400 1.70 0"711 V 2175 102.56 0"380 1.65 0"713 Cr 2130 92.42 0"377 1"60 0"908 Mn 1517 102.04 0.358 1.52 0"673

Cu 1423 89.34 0.247 1.20 0'274

Ag 1273 130.38 0.213 1.10 0"145

Au 1423 128.31 0-185 1.00 0.155

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From a perusal of figures 1-4, it is observed that the low q region is well produced for most of the elements studied. This could be adduced to two reasons, firstly, R c has been fitted to it and secondly, the Ichimaru-Utsumi form of the local field factor appears to be very good in explaining the electronic screening. For most systems studied, the

2.5

s(q)

1.5

0.5

t

^Sc

<

sCq)

<,

0 = - - ' Y " I I I I

0 2 4 6 8

q ( ~ - 1 )

2.5 i i i

1.5

1

10

0

0 2 4 6 8 10

q ( ~ - 1 )

Figure 1. Structure factor S(q) (in ~ - 1) for liquid Sc and Ti, the full curve is from HMSA theory and the points are the experimental data [13].

276 Pramana - J. Phys., Vol. 47, No. 4, October 1996

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Integral equation theory

I I l I

2 V

s(03

Figure 2.

0 2 4 6 8 10

q (.~-11 2.5

1.5 1.5

s(03

1

0.5

0

I !

Cr

0

0 2 4 6 8 10

q ()[-1) Same as in figure I but for V and Cr.

agreement between theoretical and experimental structure factors are quite good especially if one follows the philosophy of Huijben and van der Lugt [18] that for liquid metals, the locations of maxima and minima is

S(q)

are experimentally more reliable than its detailed shape. We compare our predictions for these locations in table 2. For Pramana - J. Phys., Vol. 47, No. 4, October 1996 277

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2.5 ! !

s(q)

Cu

1.5

s(q)

0

0 2 4 6 8 10

q

t I , i I

3.5

3

2.5

2

1.5

0.5

Figure 3.

0 2 4 6 8

q (~[-1)

Same as in figure 1 but for Mn and Cu.

10

all the systems investigated we obtain an accuracy of + 0.05 ~, the first peak, the only exception here is for Ti which is considerably shifted to low q region. Overall, the agreement between theory and experiment is just fair for Ti and V. F r o m table 1, one observes that for Sc, the value of the parameter a which is a measure of the hardness of 278 Pramana - J. Phys., Vol. 47, No. 4, October 1996

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Inte#ral equation theory

3 j I I I

1.5 2.5

0.5

0

2.5

0.5

0

s(q) 1.5 s(q)

Figure 4.

o 2 4 6 8 10

q (~-1)

Same as in figure ! but for Ag and Au.

0 2 4 6 $ 10

q (~1-1)

I I I I

the core gets progressively lower along the group. It is however comparatively larger for Ti and V and this could be explained on the basis that the gradient of the repulsive core part of the effective interionic potential for liquid Ti and V are relatively small compared with that of the other 3d transition metals. Ideally, one may have to build

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Table 2. Comparison of experimental and theoretical locations of the maxima and minima of the liquid structure factor for elements under consideration (in/~). The first line in each case refers to the experimental data [13] while the second line refers to the theoretical values.

Maxima Minima

Metal 1st 2nd 3rd 1st 2nd

Sc Expt. 2.50 4.80 7.40 3.55 6.00

Calc. 2-50 4-85 7.20 3.55 6.00

Ti Expt. 2.45 4.40 6-40 3.30 5.40

Calc. 2.60 5.00 7-50 3-75 6.25

V Expt. 2.70 5.00 7.20 3.70 6.20

Calc. 2.70 5.35 7-90 4.00 6.55

Cr Expt. 3'00 5.40 8'00 4.00 6'80

Calc. 2'90 5.60 8"00 4.15 6'95 Mn Expt. 2"85 5.20 7'80 3"90 6"40 Calc. 2.80 5.45 8.10 4.00 6.70 Zr Expt. 2'30 4.40 6"60 3.30 5'40 Calc. 2.30 4.50 6"65 3.30 5.50

Cu Expt. 3.00 5.40 8.00 4-00 6.80

Calc. 2-95 5.50 8.10 4.00 6.70

Ag Expt. 2.60 4.90 7.20 3.60 6.00

Calc. 2.60 4.95 7.45 3.70 6.20 Au Expt. 2"65 4.90 7.40 3"60 6.20 Calc. 2.60 4.95 7.50 3.70 6.25

into the BS potential a factor to correct for it in these systems. We note that unlike for the V M H N C calculations [11] where convergence to a solution is always obtained for all values of a, the H M S A gives consistency only for some values.

We observe that moving from 3d to the noble metals Ag and Au can be effected trivially and without difficulty using the BS potential. The agreement between theory and experiment for most of the systems studied are quite good and indicates that the HMSA which has so far proved successful for most kinds of simple liquids can be used even for noble metals without much rigour.

The results obtained using the H M S A are of comparable quality as that obtained by Bhuiyan et al [11] using the variational modified hypernetted chain approach which has a different philosophy in that it depends on the requirement that the parameter similar to ~ of the H M S A is chosen to obtain as close as possible a virial- compressibil- ity thermodynamic consistency without enforcing it.

4. Conclusion

The BS nonlocal pseudopotential has been applied successfully to calculate the structure factor of liquid 3d transition metals using the HMSA integral equation. It is found that the theory could be easily extended to noble metals in a consistent way.

280 Pramana - J. Phys., Vol. 47, No. 4, October 1996

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Integral equation theory

The calculated structure factors are good when compared with experiment. It could however be interesting to investigate the effect of using a non-local pseudopotential for Ti and V with a view to improving the result for these systems.

A c k n o w l e d g e m e n t s

One of the authors (OA) is grateful to Dr G Pastore for providing the HMSA computer programs and to the organizers of the 1995 Research Workshop on Condensed Matter Physics and for the hospitality at the International Centre for Theoretical Physics, Trieste, Italy.

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